Stability regions for maintaining efficiency in data envelopment analysis

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH European Journal of Operational Research IO8 (I 998) I27-

I39

Theory and Methodology

Stability regions for maintaining efficiency in data envelopment analysis Lawrence M. Seiford Depurtment

ofMechunical

und Industrial

Engineering.

* , Joe Zhu

ofMussuchusett.~

Unicersity

Received 7 October 1996; accepted 4

ut Amherst, Amherst,

MA 01003.

USA

February 1997

Abstract This paper develops a procedure for performing a sensitivity analysis of the efficient decision making units (DMUS) within the Chames et al. (CCR) [European Journal of Operational Research 2 (1978) 429-4441 model of data envelopment analysis (DEA). The procedure yields an exact ‘input stability region’ and ‘output stability region’ within which the efficiency of a specific efficient DMU remains unchanged. Such stability regions are simply characterized by optimal solutions of modified CCR models which are easily computed. In contrast to existing sufficient conditions for the preservation of efficiency under changes in inputs or outputs, the paper provides both necessary and sufficient conditions for an efficient DMU to remain efficient. The procedure is illustrated by numerical examples. Somewhat surprisingly, for real world data sets, for most of the efficient DMUs, the amount of some individual input can be infinitely increased when keeping other inputs and all outputs constant. This indicates that these efficient DMUs are located at extreme positions. 0 1998 Elsevier Science B.V. Keywords:

Data envelopment analysis (DEA);

Efficiency;

Sensitivity analysis; Stability

1. Introduction

In 1978, Charnes, Cooper and Rhodes (CCR) developed a mathematical programming formulation for assessing the relative efficiencies and inefficiencies of decision making units (DMUS). They termed this methodology Data Envelopment Analysis (DEA). As discussed in Seiford (19961, the evolution of DEA has been rapid and widespread resulting in a host of publications. One important issue in DEA which has been studied by many DEA researchers is the sensitivity of the results of an analysis to pertur-

bations in the data (Charnes and Neralic, 1990 and Seiford, 1994). Consider, for example, proportional increases of inputs or proportional decreases of outputs of the form ’ R;,, = P, XI,, 7 p;2

1, i= l,...,rn,

jlro = ff, Y,,, ) 0-C a,< 1, r=

l,...,

(‘1 s,

(2)

where x,,, (i= 1,2 ,..., m> and y,,, (r= 1,2 ,_.., s) are respectively, the inputs and outputs for a specific extreme efficient DMU, = DMU,,, from n DMUs,

’ Since neither an increase in output nor a decrease in input can * Corresponding

author.

[email protected].

0377-22 17/98/$19.00 P/l so377-22

Fax:

+ I-41 3-545 1027;

e-mail:

degrade an efficient DMU, direction.

0

1998 Elsevier Science B.V. All rights reserved.

17(97)00103-3

we only consider perturbations in one

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128

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Europeun

Joumul

Zhu (1996) provides a modified DEA model to compute a stability region in which DMU, remains efficient. Specifically, for an increase in inputs of form (I), this model is given by min py n s.t.

c j=

k=

l,...,m

A,Xk, I PkoXkO’ I

~

j=

foreach

AjXii

I

Xio)

i#k,

I

i:

“j Yrj 2 Yro7

r= l,...,s,

j=l A,$;

20, (3)

where x,, and Y,~ are respectively the ith input and rth output of DMUj (j = 1,. . . ,a). The approach of Zhu (1996) requires two assumptions: (i) the hyperplane constructed by the m hypothetical observations obtained from model (3) is not dominated by other DMUs besides DMU,; and (ii) model (3) is feasible. However, in real word situations, these two assumptions may not be satisfied. This paper extends the results of Zhu (1996) to develop a sensitivity analysis procedure that does not require either of these assumptions. We develop a procedure to determine an ‘input stability region (ISR)’ and an ‘output stability region (OSR)’ for a specific efficient DMU. An efficient DMU will remain efficient after the input increases of (1) or output decreases of (2) if and only if such changes occur within the ISR or OSR. It can be seen that our results also extend the approach in Charnes et al. t 1992). Note that any increase of input or any decrease of output will cause the DMUs in set E (efficient but not extreme efficient) to become inefficient. For those DMUs in set F (weakly-efficient with non-zero slacks), the amount of inputs (or outputs) which have non-zero slacks can be increased (or decreased) without limit, and these DMUs will remain in the set F. However, for inputs and outputs which have no slack, any input increase of (1) or any output decrease of (2) will cause these DMUs to become inefficient. The sensitivity issue of DMUs in set E

qfOperationu1

Research

108 (1998)

127-139

or F is straightforward if not trivial. Thus, we focus on the efficiency of the DMUs in set E, i.e., the extreme efficient DMUs. When applying our sensitivity analysis procedure to several real world data sets, we find that for most efficient DMUs, the amount of individual input can be infinitely increased while keeping other inputs and all outputs constant. This indicates that we can identify the positions of these efficient DMUs. Specifically, these efficient DMUs are located at extreme positions in the empirical production possibility set. Therefore, our procedure also provides useful managerial information in addition to the sensitivity results. The paper unfolds as follows: Section 2 develops a procedure for the sensitivity analysis of input increases under the assumption that model (3) is feasible for all individual inputs. The procedure is also applicable to decreases in outputs. In Section 3 the procedure is illustrated geometrically by a simple example. We illustrate in detail how to keep track of new generated frontier DMUs. (It may prove helpful to read Section 3 and view Fig. 1 in parallel with Section 2.) Section 4 examines the situation when model (3) (or (7) for the output case) is infeasible. In Section 5, we apply the sensitivity analysis method to several data sets in which infeasibility also occurs. Conclusions and possible extensions are summarized in Section 6.

2. A procedure for sensitivity analysis 2.1. Changes in input For DMU, E E, we first suppose that (3) is feasible for each input and consider input changes of form (1). (The case of infeasibility is discussed in Section 4). As shown in Zhu (19961, the optimal value to (3), p;*, gives the maximum possible increase for each individual input which allows DMU, to remain efficient with the other inputs and all outputs held constant. Also, (3) provides m hypothetical frontier points (efficient DMUs) when DMU,, is excluded from the empirical production possibility set. The kth point is generated by increasing the kth input from xkO to &!* xlr,, and holding all other

L.M. Seiford, J. Zhu / Europeun Journal

inputs and hypothetical

outputs constant. observations by

We denote

these

k

rif Operurional

Consider lem

the following

prob-

129

It* min C 6,” i= I

(4)

linear programming

127-139

problem where p[x,(, = xi0 + 6,” (and p, x,, = x,,, + 8, in which 0 5 6, I 8,“’ >

DMU( Pk”‘) = (x ,<,, . . . 9pil)*x/i,,,....x,,,y/,,...,~,,,).

Reseurch 108 (1998)

s.t.

i

A/x,, - 8,” IX,,,

i= l,...,m,

A,Y,, 2 Y,,, $

r= 1 ,...,s,

j=l,j#O ,,I

min C py i= I s.t.

C

A,x,,
i j=l.j#O

i= I,...,

j= I

A, L 0.

(5)

i: A,Y,, j=] _ p,“2

6,“,

m,

>Y,“,

r=

Thus, EYE, 8: is at optimality when C:‘, p,” = c:= ] p,“’ and there exist A, (j # O), s,-, ST> 0 that satisfy

l,...,s,

1, A,lO.

This model determines the smallest summation the proportions to move DMU, to the boundary the convex hull of the other DMUs.

(5’)

of of

i j-

Ajx,j+s,

=x;<,+$,,

i= l,...,m,

I

J+O

Lemma. Denote the optimal solution for (5) by p,” * (i= 1,2,. . . ,m). For i = 1,2,. . ,172,we have /3,“* 2 p:” 2 1.

2 Ajy,j j= I

Associated with py * (i = 1,2,. . . ,m), m additional points (or DMUs) can be generated as

S,? = Y,,, 7

r= 1 ,

. . ,s ,

j+O

A/,Proof. Suppose for some i,, /3,“,*< p,':,' , then CJ’,I,,fOAix,l~P~*~ ,,,” Cys,p,O* >mI +/3:* resulting in a contradiction. 0

-

,s:

2 0,

j=l

,...,n,

violating the optimality of Zy’ ,6,“*. Thus, DMU, with inputs of /?,x,~ (i = 1,. . . , m) is efficient. Note that model (5’) is useful when some of the inputs are zero. From (5’) we also note that if we apply the results in Charnes et al. (1992) to our situation of the CCR model with the input increases of form (I), then the l-norm result is equivalent to the set 0”.

DMU( p;‘) =

(x ,,,r.-.,p~*Xku,...,Xm~,,y,~,.....Yso)

(6)

Theorem 1. ForDMU,=(x,,, . . . . x,.,y~O,...~y.,O)~ denote an increase of inputs of form (1) by DMU, = (p ,,... rPm)=(P,~,o,...,Pm~mo,~,o)...,~so) and define 0” as OO=((p,,...,&)I1 Ip,Ip,~*, i = I,. . ,m) If (p,, . . . ,P,) E a”, then DMq,( p,, . . , &,I remains efficient. Proof. Suppose ( p,, . . . .&> E 0” and DMU, with inputs of fiixiO (i = 1,. . . ,m) is inefficient. In fact, (5) is equivalent to the following linear programming

Definition. Input Stability Region. A region of allowable input increases is called an Input Stabi@ Region if and only if DMU,, remains efficient after such increases occur. The input stability region (ISR) determines by how much all of DMU,‘s inputs can be increased before DMU, is within the convex hull of the other DMUs. From the lemma and Theorem 1 we know that (a) a0 is only a subset of ISR, and (b) the sets i= l,..., m, form part of the {p, 11 ~fi,
130

L.M. Se$ord. J. Zhu/Eumpean

Journal

If the input hyperplane constructed by the m points, DMU( p[ * > associated with the optimal values to (3), is not dominated by other DMUs except DMU,, i.e., that input hyperplane is a new efficient facet when excluding DMU,,, then the following set r” is precisely the ISR (Zhu, 1996)

r”=((p,...,p,)IIIP;
,...,

m and pp/3, + . . . +p,“&

where BP,. . . ,fl,” are parameters following system of equations pp*&J B; ... BP Theorem

+Ez; +p;*B; +4“;

(b) Otherwise solve model (5) for point p to obtain m new points and a similar set 0: determined by the optimal p values, say p/ * . Apply iteration t + I to each of these m new points. Stopping Rule. If the input hyperplane determined by the m points that are associated with the m optimal /? values is not dominated by other DMUs, then iteration stops.

I I},

determined

by the

+ ...

+

B; = 1

+ ...

+

B; = I

+ .‘. ..’ +

... P,“‘B,” = I.

2. In the case of input increases

of form (I), for any extreme efJicient DMU,, if the m points, DMU( p: * ), which are associated with the optimal b values to (3), determine an eflcient input hyperplane, then DMU, remains efJicient if and only if (Pi, . . . ,p,) E I-“. Proof.

Nf Opemrionul Research 108 (1998) 127-139

See Zhu (1996).

Next, suppose that the hyperplane constructed by the m points in (4) is dominated by some other DMUs which are inefficient when including DMU,. In this case the ISR is no longer the set of r”. Thus, we develop the following procedure. Initiation (t = 0). Solve model (3) for each k, k = I,. _. ,m. If the input hyperplane, which is determined by the m points of DMU( Pl* ) in (4), is not dominated by other DMUs, then we obtain the ISR defined by r”. Otherwise solve model (5). Associated with the optimal solutions to (51, pp’ , we obtain m new points, DMU( p” * ) (i = I ,2,. . . ,m> as given in (6) and nO. Iteration t = l,2, . . . ,T. At iteration t, for each point of iteration t - I, say point p, which is associated with the optimal p values to (5), we solve model (3) at each new kth input, k = 1,2,. . . ,m, and apply the Stopping Rule. (a) If the rule is satisfied for a particular point p, then we have a similar set rp’ determined by the optimal p values, say pkp * , k=l , . . . ,m. We continue for the remaining points.

From the above procedure, one can see that if a r-like set is obtained, then the iteration stops at a specific point, i.e., the r-like set indicates the termination of the iteration. Theorem

3. The input stability region is a union of 0” and some 0,: and some r,‘.

Proof. Obviously, DMU, remains efficient when its input increases ( p,, . . . , pm) belong to R” or any of the 0,: or c,‘. Conversely, from the iterations we know that the ISR is connected. Since the input increases occurred in a-like sets, DMU, is first moved to a particular point p which is used to construct a r-like set. By Theorem 2, we know that the sets of I;,’ are the boundary sets of the ISR. This means that if further input increases are not in this kind of set, then DMU, will become inefficient. Therefore, if DMU, remains efficient, then the input increase of form (I) must be in a0 or any of the 0; or 4,‘. •i

2.2. Changes in output Similarly, we can develop a sensitivity analysis procedure for output decreases of (2). For a specific extreme efficient DMU,, we consider the following linear program (Zhu, 1996) max at s.t.

k

for each k = 1,. . . ,m AjX/(jI

alY,,r

j=l,j#O

2

+rj

2

Y,o.

r#k,

j=l,j#O

i j= hj,cg

hjXi,IXi,,

i=l

,...,m,

t,J#o 2

0.

(7)

L.M. Se#wd, J. Zhu/

European Joumul

of Operutionul

Reseurch 108 (1998)

127-139

131

The optimal values to (7), (Y: * , k = 1, . . . ,s, give s hypothetical frontier points (or DMUs) and a set A” defined as follows

Theorem 5. For a decrease in outputs ofform

DMU( (Y/J’ *)

(cy I,...,

=

(x ,,,~.‘.~Xln0~ Y,o.....fff*YkC I,..‘,

CI,)Icrr”*Ia,I

I,. . . ,s and A;a,

+ . . . +A:‘a,V L 1) ,

+A;

A7

+cx;*A;

+ ...

+

“, +4

+ ...

... +

‘if

( Here we rewrite the result following theorem:

A;=

by the

=

1

... cx,,Y’*A;= 1.

of Zhu (1996)

as the

Definition. Output Stability Region. A region of allowable output decreases is called an Output Stability Region if and only if DMU,, remains efficient after such decreases occur. Now suppose that the output hyperplane constructed by the s points, DMU((Y: * >, is dominated by some other DMUs which are originally inefficient, then the output stability region (OSR) is not the set AO. We consider the following linear programming problem s max C (p: r=l 5

AjYrj 2 cPPY,0,

r= 1 ,...,S,

A,rO,

cppll.

Sx,,T

(P;*1

ko) ‘. .

Y.J .

The modification of the procedure of Section 2.1 to handle in output is straightforward. Apply model (7) and model (8) at each iteration until no A-like sets can be obtained. Similarly, we obtain Theorem 6. The output stabilig L-like sets and $-like sets.

3. Geometrical

region is a union of

illustration

We now illustrate our sensitivity analysis procedure geometrically for the following five DMUs with a single output and two inputs, For convenience, we suppose the five DMUs produce an equal amount of output and thus omit the output quantities in the following discussion. With the help of Fig. 1, we will see how to keep track of newly generated points (DMUS) by the procedure. It is obvious that DMUs 1, 2, and 3 are extreme efficient, and DMUs 4 and 5 are inefficient. Let DMU, = DMU2( xIo = 2, xzo = 21, i.e., we consider the robustness of the efficiency of DMU2 when the two inputs increase. bzitiation (t = 0). First we solve DMU, (point X,), that is

s.t. (8)

A,Xij

remains eficient.

(1 IO’. ..‘XrnD’_Y,,,,“‘,

p;’

I.j+O

2 j= I.j#O

I,...,sj

We also have the following s new points that associated with the optimal solutions, cp:‘* , of (8)

1

Proof. See Zhu (1996).

j-

Ia,

DMU( &‘* )

Theorem 4. in the case of output decreases of form (2), for any extreme efficient DMU,, if the s points, DMU( a; * ), which are associated with the optimal a values to (7), determine an eficient output hyperplane, then DMU, remains efficient if and only if ((r ,‘..‘, (Y,<)E A”.

s.t.

a,)Icp:”

(2), if

1,

in which the parameters of A: are determined following system of equations 0 * A; + .‘. + A;= “I

I,..‘, I I,r=

then DMU,

P={((Y

1, we have:

(Y,JqV={(o

,,.. . YSJ

and

r=

Similar to Theorem

=min

pp

5A, + A, + ;A, + $I, <2p;, 11 A, + SA, + $A, + Th,

i= l,...,m, A, + A, + A, + A, 2 1, A,,A,,A,,A,,P;

2 0.

52,

model

(3) for

L.M. Se@rd. J. Zhu / European Journal

132

We have pi’ * = 5/3 and for k = 2, &” = 8/5. Furthermore, we have the following two newly generated points associated with the optimal p values

B = ( xlo,&‘*

of Operutional

Research IO8 (1998)

127-139

shown in Fig. 1 and obtain the following two additional points associated with optimal p values

x2,,) =

Obviously, the input hyperplane (line segment AB) constructed by A and B is dominated by DMU4 and DMUS. Thus we solve model (5) for DMU,, that is

Irerurion (t = 1). For the first point model (3)

C, we solve

min pp + pi s.t.

5A, + A, + $A, + $A, <2p;, A, + 5h, + $A, + :As

min p,”

I 2p;,

s.t.

A, + A, + A, + A, 2 1, A,,A3,Aq,A520,p;,&2

5

AI +5A

1.

Y

5A, + A, + yh, + ,A, I /3rcc-l. x,e - 26 3

i?A24

+llA45

IX’-20

A, + A, + A, + A, 2 1, We obtain p;” = pi* = 5/4. a(’ = {( p,,&) I1 5 p, S 5/4,

Moreover, we have 1 I & I 5/4) as

A,,A3,A4,AS~0,/3,c~0.

x3

Input 1: Xl Fig. 1. Sensitivity

analysis procedure

-2,

c

*

L.M. Se@rd, J. Zhu/Europeun

Journal

We have p,“’ = 4/3. Similarly, p,” = 5/4. two corresponding new points are as follows: A=(&%& i

The

x&) = (&$).

k= 1,2and

IC,I&‘*,

a First, let x,e = d, xi = d, x,a and _?,a = d, .$, = d, p;* xzo. By Theorem 1, we have

a;=

where Bf and B: are determined

i

B:+B,C&-

i

k= 1,2}

E= B,C= ;

i

B;+;B,C=l

a

i

B;=+.

Hence,

{(d,,p;*d,)

I1 sd,

sp:*,

and further 0; = (( PI,&)1 1 I j3, s 9/8, 5/4 I & I 1 l/8). Associated with the two optimal values of pp* and pf’, we now have the following two points

as follows

;B,C+B,C= = 1 *

sd,sp,D*,

p;* 5 pz”’d, s p;*p:*},

B,cc, + B,Cc, < l),

B,CP,C* +B;=l

133

and

The input hyperplane constructed by these two points (line segment AX,) is not dominated by other DMUs, therefore the iteration for point C stops and we have the following results. Let ,?,,, = c, xF0 = c, &‘* x,e and .?,, = c2 x& = c2 xzO. By Zhu (1996), we have ~.‘=((c,,cz)Il

Reseurch IO8 (1998) 127-139

fi:,=((d,,d,)Il

,x&) = (4$2)>

x, = (x$$:*

of Operarional

i

( pp*x;,x,“o)

F= (&$*

= (x~~,x&)

x&) = (x:0.&

= (:,;), = (23.

Iteratian (t = 2). For the point E generated from the point D in the first iteration, we obtain, by solving model (31, j3: * = 10/9, & * = 11/lo, and two corresponding points

C.’ = {(c, p;” ,c*) I p;* I c, pp* < pp*p;* = p;*,

1 < c2 5 p,“,

and ipp* c,

++p;*c,rpp’). and & = c2. Then c.’ = Let p, = c, pr* 1 I Pz 5 5/4, 3P, + (@,$J5/4 I p, I5/3, 5p* < IO). Next, for the second point D, solving model (3) at k = 1 and k = 2 yields respectively pp* = 5/4 and j3p’ = 32/25. Associated with these two optimal /I values, we have two new points

The input hyperplane constructed by these two points of X, and X, is not dominated by other DMUs, therefore the iteration stops. Let ,?,, = e, ~5 = e, pp * ,rg = e, pp* x,,, and n E = e2 xfo = e2 P; * x2o. Similar to r”, we x20 = e2 x20 have r,2{(e,,e,)ll
X, = (Pp*xp,J,“,) B= (x,?,,P:*

= (&$), B,V,E’ +B;=l

XzDg)= (2,Y).

( The input hyperplane determined by these two points (X, and B) is dominated by DMUS, therefore we compute model (5) for point D, min pp + pf, s.t.

54

+ A, + $A, + $A, s p,D x,0D=2

A, + 5A, -t $A4 + $A5 I p;x&

D

PI ’ = $p;,

A, + A, + A, + A, r 1, A,,h,,A,,A,~O,pf’,p~2 We have pp* =9/8 compute 0;.

and pf’

i

_(

1

B;+#B$=

1*

B: = f i

B;=+.

Thus, ri = ((e, pp*,e, pi*> I pp’ I e, pp’ I pp’p:’ = pp’, pi* I e,p;’ < p:*p$‘*, and 9/20p/‘pi*e, + 1/2pp*p20’e,~I;~*p;“)=((P,, /3,)19/8

I p, I 5/4,

5/4

_< ,B2 I 7,

18P, + 2P2

I 45), where p, = e, pp* and p2 = e2 pp’ . For the point F, we have p[* = 9/8 and p{* = 64/55, and two corresponding points

1. = ll/lO.

B,E+B;P:*=l

$B;+B,E=

x, = (p~*x:,>x,“,) Next we i

B = (x:,&’

= ($$),

xfo) = (23).

L.M. &ford,

134

J. Zhu/European

Journal

The input hyperplane constructed by these two points of X, and B is not dominated by other DMUs, therefore the iteration stops. i,,, = f,x& = to r ‘, we have r~=((f,,f2)11 IfkI&*, k= I,2 and B,Ff, + B[f2 I 1) in which Bf and B[ are the solutions to the following system of equations Let

P,, = f,_~;

= fixlO

and

f2 p! * xg = f2 pf *pi * xzO. Similar

BK’ +fg= B:+B:P;*

= 1

Research 108 (1998)

127-139

Proof. The if part is obvious from the fact that if (3) is feasible, then the optimal value to (3) gives the maximum increase proportion of the kth input. Therefore, the amount of kth input can not be infinitely increased. To establish the only if part we suppose that the kth input is increased by M 2 1 and DMU, is inefficient. By substituting DMU, into CCR model, we have an optimal solution 8 * < 1, A,‘, A,: (j + o), in which 8 * _< 1 implies DMU, E F. Therefore,

1

$I[+B,F=

1

of Operational

B;+EB:=

’i

1

A~x,~_
j=t

Thus, r; 8/5, 9p, ing input shown in

= I( &,&)I 1 < P, I 9/8, 1 l/8 I P2 I + 5& I 17). Finally, we obtain the followstability region for DMU2 (point X,) as Fig. 1.

<

j#O

2 j=

A; x,~ I e * xi0 I xi<,,

i #

k.

I

j+O

This means that A,Y(j # o), p,, = 8 *M is a feasible solution to (3) and leads to a contradiction. Since M was arbitrary, the amount of the kth input can infinitely be increased while maintaining DMU, efficiency. Cl

4. Infeasibility The previous sensitivity analysis procedure was developed under the assumption that model (3) (or model (7)) is feasible. However, this may not be always. For instance, if we calculate (3) for DMU3 in Table 1, then we have p,* = 2 for the first input but infeasibility for the second input. Note that, in fact, we can increase infinitely the amount of DMU3’s second input while maintaining the efticiency of DMU3. Theorem 7. For an efJicient DMU, the kth

input only, model

(3)

an increase

is infeasible,

only if, the amount of kth input of increased

can be

without limitation while maintaining

efficiency of

Table

DMU,

of

if and

DMU,.

the

As the above theorem indicates, if (3) is infeasible, then p[* = +x. Thus, in this situation, we must modify the sensitivity analysis procedure, because we are unable to express the new frontier point associated with pt’ = +r. and, further, to apply the stopping rule. Note that model (5) is always feasible. ’ But in the case of infeasibility, (5) does not perform well. For instance, if we apply (5) for DMU3, we obtain p; = 2 and ~2” = 1, i.e., of* = 1 relative to the unbounded input i. Consequently, we are unable to determine the stability region. Thus, from a computational point of view, in this situation, we apply model (5) with p,O= 0, (i = 1,. . . ,m), i.e.,

I ’ Zhu (1996)

Data for the five DMUs

showed that if zeros are presented in input/output

data, then model (5) may be infeasible. However if this happens,

DMUs

1(X,)

2(X,)

3(X,)

4(X,)

5(X,)

Inputlx, Input 2 x*

5

2 2

I

5/2

5

5/2

9/4 11/4

I

we can utilize displacement change instead of proportional change and consider the non-oriented CRS model (Ali and Seiford, 1993, and Ali et al., 1995) and (5’). Therefore, we here assume that all data are positive.

L.M. S+rd,

we consider That is,

s.t.

an equal-proportional

min 0, ” C A,x,,< j=

OcJ,x,,,

J. Zhu / Europeun

Journal

increase of inputs.

i= l,...,

m,

I

/+o (9) i

‘1 Yrj L Yro

r= I,...,s,

j= I ;+o yj 2 0,

j=

1 ,...,n.

At each point, in each iteration, we first apply (9) when (3) is infeasible, and then, for the newly generated points, we apply (3). If (3) is feasible, we use the procedure suggested previously in Section 2. If (3) is still infeasible, then apply (9) again (go to next iteration). In reference to the previous procedure, we can, in fact, regard infeasibility as the rejection of the stopping rule, and then we calculate model (9) instead of (5) to generate new frontier DMUs for the next iteration. In this situation, the set 0” obtained from (9) corresponds to the z-norm in Chames et al. (1992). This general procedure for the infeasibility case is stated below: Sfep 1. Solve model (9). Step 2. Solve (3) for the newly generated by (9):

points

DMU( 6,’ ) = (x ,~~.‘..e~,*~k,,,...~x~~,,y,O,....yso)) k=l ,...,m. (a) If (3) is feasible, then go to the procedure given in Section 2; (b) If (3) is infeasible, then go to step 1. Note that infeasibility often occurs in real world situations. In theory, one can always use this general procedure to determine the ISR. However, in practice one may use this procedure to approximate the ISR due to the fact that some inputs’ amount can be infinitely augmented. For some special cases, there are some corresponding computational methods. For example, it is obvious that the IRS for DMU3 in Table 1 is 1SR=((~,,~,)Il1~,<2,1-<~~<+~).

(10)

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108 (1998)

127-139

135

which is the shaded region shown in Fig. 1. Furthermore, we have Theorem 8. For the two-input case, one of the two optimal B values in (3) is equal to the corresponding optimal value to (9), if and only if, ISR = i = 1,2}, where one of the KP,,P,)ll - 1. By 0,: and p;‘* we obtain two frontier points A = CO,,’ xlo,O~,*x2,,) and B = XJ = to,,* xlo,xJ. Thus, (Pl”‘x,,, q** = pp* 3 A E F with nonzero slack on the second input ==.( PP*x,o&xzo)

EF,

where 1 5 & < +x 2 (p,x ,,,, &x2,,) E E, where 1
of

Corollary. For the two-input case, if one of the two optimal /3 values in (3) is equal to the corresponding optimal value to (9), then (3) is infeasible for the other input. One should notice that equality is not held in the right hand inequalities in (IO) of pi. Otherwise, DMU, will be in set F. For instance, if p, = 2 in (IO), then DMU3 (X,) is moved into set F. However, if we only consider radial efficiency without slacks, i.e., weak efficiency, then the equality can be imposed, because the efficiency ratings are equal to one for the DMUs in set F.

L.M. Seiford,

136

J. Zhu/

European

Joumul

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Reseurch

I08 (1998) 127-139

Note that if (3) is infeasible, then we can determine the position of DMU,. In this situation, DMU, is located at an extreme position similar to X, or X, in Fig. 1. Therefore our sensitivity analysis procedure also provides some other useful information besides the stability. Finally, the above discussion and development holds for the output case when (7) is infeasible. In the next section, we illustrate how to implement the above procedure.

5. Applications

to some data sets

In this section we apply the sensitivity analysis procedure to sample data sets to determine the stability region. Our discussion is limited to the input-increase case. The implementation in the output case is similar. 5. I. Data set I Consider the eleven DMUs example (two inputs and two outputs) provided in Ali et al. (1995), where E = {DMUl, DMU2, DMU4, DMUS). Table 2 summarizes the sensitivity analysis results. Note that since the line segment determined by the two new points associated with each specific efficient DMU through pp * and &‘* is not dominated by the other DMUs, therefore no additional iteration is required. Note also that infeasibility occurs in one of the two inputs when solving model (3) for DMUs 2, 4 and 5. However, since the condition of Theorem 8 is satisfied, the ISRs can be easily determined.

Table 2 Sensitivity

Fig. 2. Input stability region (ISR) of DMU2.

5.2. Data set 2 Consider the actual hospital data set in Tone (1996). This data set consists of 14 DMUs (hospitals) with two inputs (number of doctors and number of nurses) and two outputs (number of outpatients and number of inpatients), where E = (DMU2, DMU3, DMU6, DMUS, DMUlO}. Infeasibility occurs in both inputs when solving (3) for DMU2. Thus, we employ (9) for the original DMU2 and for each new point generated by 0,’ from the previous iteration when infeasibility occurs, i.e., we calculate o-like sets. Otherwise, we solve (3) to determine r-like sets. The ISR for DMU2 is shown in Fig. 2 (Each dot in Fig. 2 is associated with a newly generated DMU, by 0,’ at previous iteration). That is, @ = ((/3,,&) 11 I p, < 1.05660, 1.05660), where 0,’ = 1.05660 in (9);

results for data set

1 2 P* s

I

DMU

Input stability region (ISR)

I

~(&.&)I1 < j3, < 1.63889, I I 0.42079/3, + 0.3 lO37& < I) /3,“* = 1.63889, /3;* = 1.86617

2

KP,.P*)ll

I P, < 1.5,

& < 1.86617,

1s P2< +4

PI”’ = 1.5,&’ = +r (infeasibility) ??

4

(( &.&)I I ZGP, < 1.04, 1s P2 < +=I &‘* = 1.04, &” = += (infeasibility)

5

I~P,,P*~II~P,<+~,l~P~~l.2~ &‘* = +r (infeasibility),

&‘* = I .2

~=((p,,p2)11.05660_
1 IPZS

@ = {( /3,,&) j 1.05660 I p, I 1.09236, 1.03384 5 & I 1.05660, 0.402& + 0.63185& I 1.092361, where pp* = 1.03384, /3; * = 1.02201 in (3);

@=((~,,~,)11.09236~/3,~1.11417, 1.01997}, where tI,* = 1.09117 in (9);

llP,<

L.M. Se(ford, J. Zhu/ European Journal of Operutional Research 108 (1998) 127-139

137

@={(p,,&)ll I& I 1.02127, l.O566OIP,I I .07907), where f3,>*= 1.02127 in (9);

((p,,p2)11 tain

@ = {( p,,&)[ 1.02127 I p, I 1.0566, 1.05660 5 & I 1.07907, 0.38382 p, + 0.63232& I 1.079071 where /I;‘* = 1.03459, &‘* = 1.02127 in (3); and

@={(&,&)I11/?,<+~,1r&rl.O3964}as shown in Fig. 3.

~={(&,&)I1 -<& I 1.00789, 1.079071&< 1.08758) where 0,* = 1.0789 in (9). In the remainder of the ISR, i.e., the shaded region in Fig. 2, each L&like set (when (3) is infeasible) is associated with a r-like set (boundary). /3, -+ +x when p, + 1 and p2 + +X when p, + 1. Next consider DMUlO. We have p,“’ = +x (infeasibility) and &‘* = 1.04208. Since eO*# pi*, the condition in Theorem 8 is not satisfied. Therefore, applying (9) we have 0,: = 1.03964 and two new points (0,* ~,o,~zo,ylo,y~o) and (x,,,0,* x2,,, (~,~,x~~,,y~~.y~~,) represents 1 lo3?l2,, ) 3 where DMUlO. The condition in Theorem 8 is satisfied by the first new point. Thus, we have {( p, , p2) il.03964
rp,~

1.03964=

O,,*, i=

1,2} we ob-

For the second new point, we apply the same procedure as in DMU2. The results are shown in Fig. 3, where

@=((&,&)I1 1.04166}, and

I&

I 1.00194,

1.03964<@,1

@={(/3,,&)/1 I& I 1.00194, 1.04166
set, we turn to another in Fig. 3 consists of R-like sets associated with a new DMU whose first input was increased (infeasibility), and r-like sets associated with a new DMU whose secnewly generated point. The shaded region

P*

1.04208 i 1.04166

1.03964

1.03964 Fig. 3. Input stability region (ISR) of DMUIO.

138 Table 3 Sensitivity DMU 2 3 6 8 10

L.M. Seiford, J. Zhu/Europeun

Journal

results for the data set 2

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Research IO8 (1998)

Table 4 Sensitivity

127-139

results for data set 3

Input stability region (ISR)

DMU

Input stability region (ISR)

see Fig. 2 {( &,&)I 1 < p, I 1.04190, 1 I & < 1.04113, 0.485298, +0.49438, I 1) {(&,&)I 15 /3, < 1.07520, 15 & < +I) I(P,.P~)lI
2 8

((P,,Pz,P,)Il~P,<+r,i=l,2; ((p,,&,&)Il1/3~s

1.36809, i=

11&<1.28002) 1.2; 15 p,< +x1

{(PI,&,&)1 15 Pj I 1.21788, i = 2.3; 1.36809 I p, < 1.66617) (( &,&,&)I I < Pi I 1.21788, i = 1,3; 1.36809 5 & I 1.74689)

((P,.&,P,)Il

5 Pjsl1.19938,

i = 2.3; 1.66617 I PI < 1.99837)

and so on, until p, + +x when p, + 1 (i = 2.3) and & + + x when p, + 1 (i = 1.3)

ond input was increased (feasibility). In the shaded region, p, -+ + = when & -+ 1.03964. (See Table 3.) 5.3. Data set 3 Consider the real world data set for the Chinese iron and steel industry in 1989 in Seiford et al. (1996). This data set consists of 10 DMUs (major iron and steel firms), where E = (DMU2, DMU8). There are three inputs of labor, working capital and fixed capital, and two outputs of crude steel and pig iron. For DMU2, we have BP* = BP* = + 3f (infeasible) and p;’ = 1.28024 and 0,,* = 1.26245 in (9). However, we have 0,* = 1.26245 at each new point which is generated by increasing the first and the second inputs of a point the previous iteration. In addition, we first have 0,’ = 1.0125 and then 0,* = 1.00140 at each new point which is generated by increasing the third input of a point from the previous iteration. Note that the product of 1.26245, 1.0125 and 1.00140 is 1.28002 which is very close to p;* = 1.28024. Hence we can summarize the sensitivity results as reported in Table 4. ForDMU8, we have p;* = &‘* = #* = +x.So we apply the general procedure described in the previous section. The results are given in the Table 4. It can be seen that sometimes when infeasibility occurs, we can only approximate the ISR, e.g., DMU2 and DMUlO in data set 2, and DMU2 and DMU8 in data set 3, since some inputs are unbounded However our procedure can yield an ISR as large as desired. This is useful in examining the stability of an efficient DMU for which some input increases are unbounded. It can also be seen from the above applications

that the amount of individual input for most DMUs can be infinitely increased when other inputs and all outputs constant. This that most efficient DMUs are located at the like X, or X, in Fig. 1, while the X,-like less frequently occur.

6. Conclusions

efficient keeping indicates positions positions

and possible extensions

The current paper has provided a procedure for the sensitivity analysis of the efficient DMUs in the CCR model and has extended Zhu’s approach thereby making it applicable to more general situations (Zhu, 1996). An input stability region (ISR) and an output stability region (OSR) are calculated respectively for possible increases in all inputs and for possible decreases in all outputs in order for a DMU to

Table 5 Data set 1 DMU

Input 1

Input 2

output

1 2 3 4 5 6 I 8 9 10 11

40 30 93 50 80 35 105 97 100 90 98

30 60 40 70 30 45 75 67 50 60 65

160 180 170 190 180 140 120 100 140 140 140

Source: Ali et al. (1995).

1

output 2 100 70 60 130 120 82 90 82 40 105 50

L.M. Seiford, J. Zhu/European

Journal

Table 6 Data set 2 DMU I 2 3 4 5 6 7 8 9

I0 II I2 13 I4

Input 1

Input 2

output

3008 3985 4324 3534 8836 5376 4982 4775 8046 8554 6147 8366 I3479 21808

20980 25643 26978 25361 40796 37562 33088 39122 42958 48955 45514 55140 68037 78302

97775 135871 133655 46243 176661 182576 98880 136701 225138 257370 165274 203989 I74270 322990

1

output

2

101225 130580 168473 IO0407 215616 217615 I67278 193393 256575 312877 227099 321623 34 1743 487539

I 2 3 4 5 6 7 8 9 IO

182 74 160 183 133 106 109 240 276 191

Input 2

Input 3

output

237 82 I95 150 I55 120 I10 243 188 117

468 148 400 339 329 138 188 806 574 466

5008 1857 4041 2779 3506 I306 1515 7763 4577 3322

I

output

139

We provide the three data sets cited in Section 5 (Tables 5-7).

Table 7 Data set 3

1

127-139

Appendix A. Data sets

remain efficient. A unique feature of this procedure is that it gives necessary and sufficient conditions for an efficient DMU to maintain efficiency when all inputs and all outputs change respectively. Additional information is provided by the sensitivity analysis procedure where infeasibility can characterize the positions of efficient DMUs. Note also that one can employ the technique of Zhu (1996) to consider the simultaneous change of all inputs (1) and change of all outputs (2), but the necessary condition for preserving efficiency no

Input

Research 108 (1998)

longer holds. Thus, further research should be focused on developing such a necessary condition. Other possible extensions currently under investigation by the authors include (a) the simultaneous change in all DMUs’ data, including both efficient DMUs and inefficient DMUs; (b) sensitivity analysis for other DEA models (Charnes et al., 1994); and, (c) the relationship between infeasibility and stability.

Source: Tone (I 996). Inputs I and 2 are numbers of doctors and nurses, respectively. Outputs 1 and 2 are numbers of outpatients and inpatients, respectively.

MU

of Operational

2

5303 2336 5001 2418 3602 956 2282 9601 6493 4233

Source: Seiford and Zhu (1996). Inputs I, 2 and 3 are amounts of labor, working capital and fixed capital utilized. Outputs 1 and 2 are amounts of crude steel and pig iron produced.

References Ali,

A.I., Lerme, C.S., Seiford, L.M., 1995. Components of efficiency evaluation in data envelopment analysis. European Journal of Operational Research 80, 462-473. Ali, A.I., Seiford, L.M., 1993. The mathematical programming approach to efficiency measurement. In: Fried, H., Lovell, C.A.K., Schmidt, S. @Is..), The Measurement of Productive Efficiency: Techniques and Applications. Oxford University Press. London. Chames, A., Cooper, W.W., Rhodes, E., 1978. Measuring the efficiency of decision making units. European Journal of Operational Research 2 (6). 429-444 Chames, A., Neralic, L., 1990. Sensitivity analysis of the additive model in data envelopment analysis. European Journal of Operational Research 48, 332-341. Chames, A., Haag, S., Jaska, P., Semple, J., 1992. Sensitivity of efficiency classifications in the additive model of data envelopment analysis. Int. Journal System Science 23 (5). 789-798. Charnes, A., Cooper, W.W., Lewin, A., Seiford, L.M., 1994. Data Envelopment Analysis: Theory, Methodology and Applications. Kluwer Academic Publishers. Seiford, L.M., 1994. A DEA bibliography. In: Chames, A., Cooper, WW. Lewin, A., Seiford, L.M., Data Envelopment Analysis: Theory, Methodology and Applications. Kluwer Academic Publishers. Seiford, L.M., 1996. Data envelopment analysis: The evolution of the state of the art (1978- 1995). Journal of Productivity Analysis 7, 99- 138. Seiford, L.M., Zhu, J., 1996. Market entity behavior of Chinese state-owned enterprises: working paper. Tone, K., 1996. A simple characterization of returns to scale in DEA. Research Report. Journal of the Operations Research Society of Japan, Vol. 39, No. 4. Zhu, J., 1996. Robustness of the efficient DMUs in data envelopment analysis. European Journal of Operational Research 90, 45 I-460.